That C6 manual is scarce on the bits of info that are really needed, like what pin works what relay.
The FC250 wiring diagram you've linked to shows the use of an additional DPDT (Double Pole Double Throw) relay to change direction, along with a DPDT switch to select manual or CNC speed control, but will still rely on you having the main switch set to forward for the spindle to actually power up. One thing I notice with that set-up, is the spindle power board would be on while changing direction, which has me wondering if that would be healthy for the power board.
Do you still want to keep manual control?
If not, I'd be inclined to bypass the direction switch, add a DPDT relay to control direction via one of the relays on the C6, and use the other C6 relay to enable the spindle power board.